Random paths with bounded local time

TL;DR

This paper studies a conditioned Brownian motion with bounded local time, revealing entropic repulsion that makes the process ballistic with a specific asymptotic velocity, and extends the analysis to random walks.

Contribution

It introduces a new conditioned process with bounded local time, demonstrating entropic repulsion and ballistic behavior, and provides explicit velocity characterization for Brownian motion.

Findings

Brownian motion exhibits entropic repulsion under bounded local time conditioning

The process is ballistic with an asymptotic velocity around 4.58 for Brownian motion

Random walk case also shows ballisticity, but with an unknown speed

Abstract

We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process is asymptotically ballistic but with an unknown speed.